1. An Introduction to Cryptology and Coding Theory Discrete Math 2006

2. Communication System Digital SourceDigital Sink Source Encoding Source Decoding EncryptionDecryption Error Control Encoding Error Control Decoding ModulationChannelDemodulation

3. Cryptology Cryptography  Inventing cipher systems; protecting communications and storage Cryptanalysis  Breaking cipher systems

4. Cryptography

5. Cryptanalysis

6. What is used in Cryptology? Cryptography:  Linear algebra, abstract algebra, number theory Cryptanalysis:  Probability, statistics, combinatorics, computing

7. Caesar Cipher ABCDEFGHIJKLMNOPQRSTUVWXYZ Key = 3 DEFGHIJKLMNOPQRSTUVWXYZABC Example  Plaintext: OLINCOLLEGE  Encryption: Shift by KEY = 3  Ciphertext: ROLQFROOHJH  Decryption: Shift backwards by KEY = 3

8. Cryptanalysis of Caesar Try all 26 possible shifts Frequency analysis

9. Substitution Cipher Permute A-Z randomly: A B C D E F G H I J K L M N O P… becomes H Q A W I N F T E B X S F O P C… Substitute H for A, Q for B, etc. Example  Plaintext: OLINCOLLEGE  Key: PSEOAPSSIFI

10. Cryptanalysis of Substitution Ciphers Try all 26! permutations (?) Frequency analysis

11. One-Time Pads Map A, B, C, … Z to 0, 1, 2, …25 Plaintext: MATHISUSEFULANDFUN Key: NGUJKAMOCTLNYBCIAZ Encryption: “Add” key to message mod 26 Decryption: “Subtract” key from ciphertext mod 26

12. One-Time Pads Unconditionally secure Problem: Exchanging the key There are some clever ways to exchange the key….

13. Public-Key Cryptography Diffie & Hellman (1976) Known at GCHQ years before Uses one-way (asymmetric) functions, public keys, and private keys

14. Public Key Algorithms Based on two hard problems  Factoring large integers (Duc and Andrew)  The discrete logarithm problem

15. WWII Folly: The Weather- Beaten Enigma

16. Need more than secrecy…. Need reliability! Enter coding theory…..

17. What is Coding Theory? Coding theory is the study of error- control codes Error control codes are used to detect and correct errors that occur when data are transferred or stored

18. What IS Coding Theory? A mix of mathematics, computer science, electrical engineering, telecommunications  Linear algebra  Abstract algebra (groups, rings, fields)  Probability&Statistics  Signals&Systems  Implementation issues  Optimization issues  Performance issues

19. General Problem We want to send data from one place to another…  channels: telephone lines, internet cables, fiber-optic lines, microwave radio channels, cell phone channels, etc. or we want to write and later retrieve data…  channels: hard drives, disks, CD-ROMs, DVDs, solid state memory, etc. BUT! the data, or signals, may be corrupted  additive noise, attenuation, interference, jamming, hardware malfunction, etc.

20. General Solution Add controlled redundancy to the message to improve the chances of being able to recover the original message Trivial example: The telephone game

21. How Good Does It Get? What are the ideal trade-offs between rate, error-correcting capability, and number of codewords? What is the biggest distance you can get given a fixed rate or fixed number of codewords? What is the best rate you can get given a fixed distance or fixed number of codewords?

22. Who Cares? You and me!  Shopping and e-commerce  ATMs and online banking  Satellite TV & Radio, Cable TV, CD players  Corporate/government espionage Who else?  NSA, IDA, RSA, Aerospace, Bell Labs, AT&T, NASA, Lucent, Amazon, iTunes…